Journal articles: 'Invariants of knots and 3-manifolds'

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Morton, H. R. "QUANTUM INVARIANTS OF KNOTS AND 3‐MANIFOLDS." Bulletin of the London Mathematical Society 28, no. 6 (November 1996): 669–70. http://dx.doi.org/10.1112/blms/28.6.669.

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CHERNOV TCHERNOV, VLADIMIR. "FRAMED KNOTS IN 3-MANIFOLDS AND AFFINE SELF-LINKING NUMBERS." Journal of Knot Theory and Its Ramifications 14, no. 06 (September 2005): 791–818. http://dx.doi.org/10.1142/s0218216505004056.

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The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S3 is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds. As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S1 × S2)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S1 × S2)# M′ we construct K in M such that |K| = 2 ≠ ∞.

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Kalfagianni, Efstratia. "Finite type invariants for knots in 3-manifolds." Topology 37, no. 3 (May 1998): 673–707. http://dx.doi.org/10.1016/s0040-9383(97)00034-7.

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KAWAUCHI, AKIO. "ON LINKING SIGNATURE INVARIANTS OF SURFACE-KNOTS." Journal of Knot Theory and Its Ramifications 11, no. 03 (May 2002): 369–85. http://dx.doi.org/10.1142/s0218216502001688.

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We show that the linking signature of a closed oriented 4-manifold with infinite cyclic first homology is twice the Rochlin invariant of an exact leaf with a spin support if such a leaf exists. In particular, the linking signature of a surface-knot in the 4-sphere is twice the Rochlin invariant of an exact leaf of an associated closed spin 4-manifold with infinite cyclic first homology. As an application, we characterize a difference between the spin structures on a homology quaternion space in terms of closed oriented 4-manifolds with infinite cyclic first homology, so that we can obtain examples showing that some different punctured embeddings into S4 produce different Rochlin invariants for some homology quaternion spaces.

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MIZUSAWA, ATSUHIKO, and JUN MURAKAMI. "INVARIANTS OF HANDLEBODY-KNOTS VIA YOKOTA'S INVARIANTS." Journal of Knot Theory and Its Ramifications 22, no. 11 (October 2013): 1350068. http://dx.doi.org/10.1142/s0218216513500685.

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We construct quantum [Formula: see text] type invariants for handlebody-knots in the 3-sphere S3. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten–Reshetikhin–Turaev invariants.

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Kauffman, Louis Hirsch, and Vassily Olegovich Manturov. "Graphical constructions for the sl(3), C2 and G2 invariants for virtual knots, virtual braids and free knots." Journal of Knot Theory and Its Ramifications 24, no. 06 (May 2015): 1550031. http://dx.doi.org/10.1142/s0218216515500315.

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We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) and G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.

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Kuperberg, Greg. "Book Review: Quantum invariants of knots and 3-manifolds." Bulletin of the American Mathematical Society 33, no. 01 (January 1, 1996): 107–11. http://dx.doi.org/10.1090/s0273-0979-96-00621-0.

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Nikshych, Dmitri, Vladimir Turaev, and Leonid Vainerman. "Invariants of knots and 3-manifolds from quantum groupoids." Topology and its Applications 127, no. 1-2 (January 2003): 91–123. http://dx.doi.org/10.1016/s0166-8641(02)00055-x.

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Cahn, Patricia, Vladimir Chernov, and Rustam Sadykov. "The number of framings of a knot in a 3-manifold." Journal of Knot Theory and Its Ramifications 23, no. 13 (November 2014): 1450072. http://dx.doi.org/10.1142/s0218216514500722.

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In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.

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Gukov, Sergei, Du Pei, Pavel Putrov, and Cumrun Vafa. "BPS spectra and 3-manifold invariants." Journal of Knot Theory and Its Ramifications 29, no. 02 (February 2020): 2040003. http://dx.doi.org/10.1142/s0218216520400039.

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We provide a physical definition of new homological invariants [Formula: see text] of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on [Formula: see text] times a 2-disk, [Formula: see text], whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d [Formula: see text] theory [Formula: see text]: [Formula: see text] half-index, [Formula: see text] superconformal index, and [Formula: see text] topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of [Formula: see text]. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.

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YAMAMOTO, MINORU. "FIRST ORDER SEMI-LOCAL INVARIANTS OF STABLE MAPS OF 3-MANIFOLDS INTO THE PLANE." Proceedings of the London Mathematical Society 92, no. 2 (February 20, 2006): 471–504. http://dx.doi.org/10.1112/s0024611505015534.

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In the late 1980s, Vassiliev introduced new graded numerical invariants of knots, which are now called Vassiliev invariants or finite-type invariants. Since he made this definition, many people have been trying to construct Vassiliev type invariants for various mapping spaces. In the early 1990s, Arnold and Goryunov introduced the notion of first order (local) invariants of stable maps. In this paper, we define and study {\it first order semi-local invariants} of stable maps and those of stable fold maps of a closed orientable 3-dimensional manifold into the plane. We show that there are essentially eight first order semi-local invariants. For a stable map, one of them is a constant invariant, six of them count the number of singular fibers of a given type which appear discretely (there are exactly six types of such singular fibers), and the last one is the Euler characteristic of the Stein factorization of this stable map. Besides these invariants, for stable fold maps, the Bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi-local invariant. Our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps.

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LI, WEIPING, and QINGXUE WANG. "AN SL2(ℂ) ALGEBRO-GEOMETRIC INVARIANT OF KNOTS." International Journal of Mathematics 22, no. 09 (September 2011): 1209–30. http://dx.doi.org/10.1142/s0129167x11007240.

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In this paper, we define a new algebro-geometric invariant of three-manifolds resulting from Dehn surgery along a hyperbolic knot complement in S3. We establish a Casson-type invariant for these three-manifolds. In the last section, we explicitly calculate the character variety of the figure-eight knot and discuss some applications, as well as the computation of our new invariants for some three-manifolds resulting from Dehn surgery along the figure-eight knot.

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Vance, Katherine. "Tau invariants for balanced spatial graphs." Journal of Knot Theory and Its Ramifications 29, no. 09 (August 2020): 2050066. http://dx.doi.org/10.1142/s0218216520500662.

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In 2003, Ozsváth and Szabó defined the concordance invariant [Formula: see text] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [Formula: see text] for knots in [Formula: see text] and a combinatorial proof that [Formula: see text] gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in [Formula: see text], extending HFK for knots. We define a [Formula: see text]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [Formula: see text] invariant for balanced spatial graphs generalizing the [Formula: see text] knot concordance invariant. In particular, this defines a [Formula: see text] invariant for links in [Formula: see text]. Using techniques similar to those of Sarkar, we show that our [Formula: see text] invariant is an obstruction to a link being slice.

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IWAKIRI, MASAHIDE. "CALCULATION OF DIHEDRAL QUANDLE COCYCLE INVARIANTS OF TWIST SPUN 2-BRIDGE KNOTS." Journal of Knot Theory and Its Ramifications 14, no. 02 (March 2005): 217–29. http://dx.doi.org/10.1142/s0218216505003798.

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Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τrT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τrS(α,β) by a similar method.

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FENN, ROGER, and COLIN ROURKE. "RACKS AND LINKS IN CODIMENSION TWO." Journal of Knot Theory and Its Ramifications 01, no. 04 (December 1992): 343–406. http://dx.doi.org/10.1142/s0218216592000203.

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A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

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Taylor, Scott, and Maggy Tomova. "Additive invariants for knots, links and graphs in 3–manifolds." Geometry & Topology 22, no. 6 (September 23, 2018): 3235–86. http://dx.doi.org/10.2140/gt.2018.22.3235.

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17

KODA, YUYA. "LINKS AND SPINES." Journal of Knot Theory and Its Ramifications 21, no. 03 (March 2012): 1250027. http://dx.doi.org/10.1142/s0218216511009674.

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We define two kinds of invariants of links in closed 3-manifolds, the s-complexity(s ∈ ℕ) and the block number, by considering decompositions of links in closed orientable 3-manifolds by spines. The first one is a generalization of the complexity of links defined by Pervova and Petronio. After providing properties of these invariants, we construct special spines of strongly-cyclic coverings branched over generalized twist knots in lens spaces, including S3 and ℝP3, which provide upper bounds for the invariants.

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SAKURAI, MIGIWA. "2- AND 3-VARIATIONS AND FINITE TYPE INVARIANTS OF DEGREE 2 AND 3." Journal of Knot Theory and Its Ramifications 22, no. 08 (July 2013): 1350042. http://dx.doi.org/10.1142/s0218216513500429.

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Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

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Bellettini, Giovanni, Maurizio Paolini, and Yi-Sheng Wang. "A complete invariant for connected surfaces in the 3-sphere." Journal of Knot Theory and Its Ramifications 29, no. 01 (January 2020): 1950091. http://dx.doi.org/10.1142/s0218216519500913.

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We construct a complete invariant of oriented connected closed surfaces in [Formula: see text], which generalizes the notion of peripheral system of a knot group. As an application, we define two computable invariants to investigate handlebody knots and bi-knotted surfaces with homeomorphic complements. In particular, we obtain an alternative proof of inequivalence of Ishii, Kishimoto, Moriuchi and Suzuki’s handlebody knots [Formula: see text] and [Formula: see text].

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20

NAIK, SWATEE. "New invariants of periodic knots." Mathematical Proceedings of the Cambridge Philosophical Society 122, no. 2 (September 1997): 281–90. http://dx.doi.org/10.1017/s0305004197001801.

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For knots in S3 we obtain new criteria for periodicity. We show that the Casson–Gordon invariants of a periodic knot are preserved under the periodic action lifted to the cyclic covers. As an application, we consider a family of knots with the Seifert form of a period 3 knot, and using Casson–Gordon invariants show that knots in this family do not have period 3. We also obtain periodicity criteria in terms of the homology groups of cyclic branched covers of S3.

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Marengon, Marco. "On d-invariants and generalized Kanenobu knots." Journal of Knot Theory and Its Ramifications 25, no. 08 (July 2016): 1650048. http://dx.doi.org/10.1142/s0218216516500486.

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We prove that for particular infinite families of [Formula: see text]-spaces, arising as branched double covers, the [Formula: see text]-invariants defined by Ozsváth and Szabó assume arbitrarily large positive and negative values. As a consequence, we generalize a result by Greene and Watson by proving, for every odd number [Formula: see text], the existence of infinitely many non-quasi-alternating homologically thin knots with determinant [Formula: see text], and a result by Hoffman and Walsh concerning the existence of hyperbolic weight [Formula: see text] manifolds, that are not surgery on a knot in [Formula: see text].

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GAROUFALIDIS, STAVROS, and THANG TQ LÊ. "IS THE JONES POLYNOMIAL OF A KNOT REALLY A POLYNOMIAL?" Journal of Knot Theory and Its Ramifications 15, no. 08 (October 2006): 983–1000. http://dx.doi.org/10.1142/s0218216506004919.

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The Jones polynomial of a knot in 3-space is a Laurent polynomial in q, with integer coefficients. Many people have pondered why this is so, and what a proper generalization of the Jones polynomial for knots in other closed 3-manifolds is. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we argue that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.

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Boden, Hans U., and Cynthia L. Curtis. "The SL(2, C) Casson Invariant for Knots and the Â-polynomial." Canadian Journal of Mathematics 68, no. 1 (February 1, 2016): 3–23. http://dx.doi.org/10.4153/cjm-2015-025-5.

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AbstractIn this paper, we extend the definition of the SL(2,ℂ) Casson invariant to arbitrary knots K in integral homology 3-spheres and relate it to the m-degree of the Â-polynomial of K. We prove a product formula for the Â-polynomial of the connected sum K1#K2 of two knots in S3 and deduce additivity of the SL(2,ℂ) Casson knot invariant under connected sums for a large class of knots in S3. We also present an example of a nontrivial knot K in S3 with trivial Â-polynomial and trivial SL(2,ℂ) Casson knot invariant, showing that neither of these invariants detect the unknot.

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Andersen, Jørgen Ellegaard, and Søren Fuglede Jørgensen. "On the Witten–Reshetikhin–Turaev invariants of torus bundles." Journal of Knot Theory and Its Ramifications 24, no. 11 (October 2015): 1550055. http://dx.doi.org/10.1142/s0218216515500558.

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By methods similar to those used by L. Jeffrey [L. C. Jeffrey, Chern–Simons–Witten invariants of lens spaces and torus bundles, and the semiclassical approximation, Commun. Math. Phys.147 (1992) 563–604], we compute the quantum SU (N)-invariants for mapping tori of trace 2 homeomorphisms of a genus 1 surface when N = 2, 3 and discuss their asymptotics. In particular, we obtain directly a proof of a version of Witten's asymptotic expansion conjecture for these 3-manifolds. We further prove the growth rate conjecture for these 3-manifolds in the SU(2) case, where we also allow the 3-manifolds to contain certain knots. In this case we also discuss trace -2 homeomorphisms, obtaining — in combination with Jeffrey's results — a proof of the asymptotic expansion conjecture for all torus bundles.

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Lickorish, W. B. R. "Sampling the SU(N) Invariants of Three-Manifolds." Journal of Knot Theory and Its Ramifications 06, no. 01 (February 1997): 45–60. http://dx.doi.org/10.1142/s0218216597000054.

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The SU(N) quantum invariants for three-manifolds have been established in a combinatorial way by Yokota starting from the skein theory associated with the HOMFLY polynomial invariant of knot theory. Using Yokota's formulation it is here noted that there are distinct three-manifolds that are not distinguished by these invariants and that there are manifolds distinguished by the SU(N) invariants for N ≥ 3 that have the same SU(2) invariants. A by-product of the investigation is a reversing result for the HOMFLY polynomial.

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GRISHANOV, S. A., V. R. MESHKOV, and V. A. VASSILIEV. "RECOGNIZING TEXTILE STRUCTURES BY FINITE TYPE KNOT INVARIANTS." Journal of Knot Theory and Its Ramifications 18, no. 02 (February 2009): 209–35. http://dx.doi.org/10.1142/s0218216509006902.

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Murakami, Jun. "Generalized Kashaev invariants for knots in three manifolds." Quantum Topology 8, no. 1 (2017): 35–73. http://dx.doi.org/10.4171/qt/86.

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ZENKINA, M. V. "THE PARITY HIERARCHY AND NEW INVARIANTS OF KNOTS IN THICKENED SURFACES." Journal of Knot Theory and Its Ramifications 22, no. 04 (April 2013): 1340001. http://dx.doi.org/10.1142/s0218216513400014.

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In this paper, we construct an invariant for virtual knots in the thickened sphere Sg with g handles; this invariant is a Laurent polynomial in 2g + 3 variables. To this end, we use a modification of the Wirtinger presentation of the knot group and the concept of parity introduced by Manturov. By using this invariant, one can prove that the knots shown in Fig. 1 are not equivalent [S. A. Grishanov, V. R. Meshkov and V. A. Vassiliev, Recognizing textile structures by finite type knot invariants, J. Knot Theory Ramifications18(2) (2009) 209–235]. Section 4 of the paper is devoted to an enhancement of the invariant (construction of the invariant module) by using the parity hierarchy concept suggested by Manturov. Namely, we discriminate between odd crossings and two types of even crossings; the latter two types depend on whether an even crossing remains even/odd after all odd crossings of the diagram are removed. The construction of the invariant also works for virtual knots.

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Beliakova, Anna, and Thang T. Q. Lê. "Integrality of quantum 3-manifold invariants and a rational surgery formula." Compositio Mathematica 143, no. 6 (November 2007): 1593–612. http://dx.doi.org/10.1112/s0010437x07003053.

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AbstractWe prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

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FUJII, HIROZUMI. "FIRST COMMON TERMS OF THE HOMFLY AND KAUFFMAN POLYNOMIALS, AND THE CONWAY POLYNOMIAL OF A KNOT." Journal of Knot Theory and Its Ramifications 08, no. 04 (June 1999): 447–62. http://dx.doi.org/10.1142/s0218216599000316.

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We study the first common terms of the HOMFLY and Kauffman polynomials of a knot, which we call the ϒ-polynomial, and the Conway polynomial for the 2-bridge knots and a class of 3-bridge knots. We characterize the ϒ-polynomial using the 2-bridge knots. Then we give some relations between the two polynomial invariants, and as an application, we consider the space of the Vassiliev invariant of order four.

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Flapan, Erica. "Infinitely Periodic Knots." Canadian Journal of Mathematics 37, no. 1 (February 1, 1985): 17–28. http://dx.doi.org/10.4153/cjm-1985-002-4.

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One aspect of the study of 3-manifolds is to determine what finite group actions a given manifold has. Some important questions that one can ask about these actions on a given manifold are: What periods could they have? and, what sets of points may be fixed by the action? In the case of periodic transformations of homology spheres, Smith [18] classified the types of fixed point sets which could occur. For homology 3-spheres the fixed point set will be ∅, S0, S1, or S2. Fox [4] looked at periodic transformations of the three sphere which leave a knot invariant and, using Smith's classification of fixed point sets, determined that there were eight types of transformations according to how the fixed point set met the knot. For convenience we shall say a knot is (a, b)-periodic if there is a periodic transformation of S3 leaving the knot invariant with fixed point set homeomorphic to a and with the fixed point set meeting the knot in a set homeomorphic to b.

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Christensen, Antje. "Homology of manifolds obtained by Dehn surgery on knots in lens spaces." Journal of Knot Theory and Its Ramifications 09, no. 04 (June 2000): 431–42. http://dx.doi.org/10.1142/s0218216500000219.

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The question whether or not a Dehn surgery on a knot in a lens space yields a lens space of the same order is investigated with homological techniques. Determining the first homology group of the lens space after surgery and of its covering yields some necessary conditions on the knot and the surgery curve. Application of these results along with a calculation of Seifert invariants answers the question completely for surgery on torus knots along nullhomological curves.

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Manion, Andrew. "The rational Khovanov homology of 3-strand pretzel links." Journal of Knot Theory and Its Ramifications 23, no. 08 (July 2014): 1450040. http://dx.doi.org/10.1142/s0218216514500400.

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The 3-strand pretzel knots and links are a well-studied source of examples in knot theory. However, while there have been computations of the Khovanov homology and Rasmussen s-invariants of some sub-families of 3-strand pretzel knots, no general formula has been given for all of them. We give a formula for the unreduced Khovanov homology, over the rational numbers, of all 3-strand pretzel links. We also compute generalized s-invariants of these links.

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Kirk, Paul, and Charles Livingston. "Type 1 knot invariants in 3-manifolds." Pacific Journal of Mathematics 183, no. 2 (April 1, 1998): 305–31. http://dx.doi.org/10.2140/pjm.1998.183.305.

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Ito, Noboru, and Yusuke Takimura. "Crosscap number of knots and volume bounds." International Journal of Mathematics 31, no. 13 (November 28, 2020): 2050111. http://dx.doi.org/10.1142/s0129167x20501116.

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In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

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Heck, Prudence. "Twisted Homology Cobordism Invariants of Knots in Aspherical Manifolds." International Mathematics Research Notices 2012, no. 15 (August 8, 2011): 3434–82. http://dx.doi.org/10.1093/imrn/rnr145.

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GAROUFALIDIS, STAVROS, and JEROME LEVINE. "On finite type 3-manifold invariants IV: comparison of definitions." Mathematical Proceedings of the Cambridge Philosophical Society 122, no. 2 (September 1997): 291–300. http://dx.doi.org/10.1017/s0305004196001235.

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The present paper is a continuation of [Ga], [GL1] and [GO]. Using a key lemma we compare two currently existing definitions of finite type invariants of oriented integral homology spheres and show that type 3m invariants in the sense of Ohtsuki [Oh] are included in type m invariants in the sense of the first author [Ga]. This partially answers question 1 of [Ga]. We show that type 3m invariants of integral homology spheres in the sense of Ohtsuki map to type 2m invariants of knots in S3, thus answering question 2 from [Ga].

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38

BAADHIO, RANDY A., and LOUIS H. KAUFFMAN. "LINK MANIFOLDS AND GLOBAL GRAVITATIONAL ANOMALIES." Reviews in Mathematical Physics 05, no. 02 (June 1993): 331–43. http://dx.doi.org/10.1142/s0129055x93000085.

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A new method of constructing exotic 11-dimensional structures, as solutions for the cancellation of global gravitational anomalies, is revealed. The construction makes use of link manifolds. Of relevant interest is the fact that the geometry and topology of the exotic 11-dimensional manifolds are encoded in the classical invariants of knots and links in dimension three.

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Grishanov, S. A., and V. A. Vassiliev. "Two constructions of weight systems for invariants of knots in non-trivial 3-manifolds." Topology and its Applications 155, no. 16 (October 2008): 1757–65. http://dx.doi.org/10.1016/j.topol.2007.11.006.

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Kirk, Paul, and Charles Livingston. "Knot invariants in 3-manifolds and essential tori." Pacific Journal of Mathematics 197, no. 1 (January 1, 2001): 73–96. http://dx.doi.org/10.2140/pjm.2001.197.73.

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41

Kholodenko, Arkady. "Black magic session of concordance: Regge mass spectrum from Casson’s invariant." International Journal of Modern Physics A 30, no. 33 (November 26, 2015): 1550189. http://dx.doi.org/10.1142/s0217751x15501894.

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Recently, there had been a great deal of interest in obtaining and describing all kinds of knots in links in hydrodynamics, electrodynamics, non-Abelian gauge field theories and gravity. Although knots and links are observables of the Chern–Simons (CS) functional, the dynamical conditions for their generation lie outside the scope of the CS theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S3are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space is thus far attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of nonnegative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of nonnegative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is a specialization of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson’s surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew–Frautschi (CF) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding CF plots. The physics behind Casson’s surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons.

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Meyerhoff, Robert, and Mingqing Ouyang. "The η-Invariants of Cusped Hyperbolic 3-Manifolds." Canadian Mathematical Bulletin 40, no. 2 (June 1, 1997): 204–13. http://dx.doi.org/10.4153/cmb-1997-025-8.

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AbstractIn this paper, we define the η-invariant for a cusped hyperbolic 3-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

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43

Kotorii, Yuka. "A relation between Milnor’s μ-invariants and HOMFLYPT polynomials." Journal of Knot Theory and Its Ramifications 25, no. 13 (November 2016): 1650072. http://dx.doi.org/10.1142/s0218216516500723.

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Polyak showed that any Milnor’s [Formula: see text]-invariant of length 3 can be represented as a combination of the Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called [Formula: see text]-invariants. We show that any Milnor’s [Formula: see text]-invariant of length [Formula: see text] can be represented as a combination of the HOMFLYPT polynomials of knots obtained from the string link by some operation, if all [Formula: see text]-invariants of length [Formula: see text] vanish. Moreover, [Formula: see text]-invariants of length [Formula: see text] are given by a combination of the Conway polynomials and linking numbers without any vanishing assumption.

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44

TURAEV, VLADIMIR G. "MODULAR CATEGORIES AND 3-MANIFOLD INVARIANTS." International Journal of Modern Physics B 06, no. 11n12 (June 1992): 1807–24. http://dx.doi.org/10.1142/s0217979292000876.

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The aim of this paper is to give a concise introduction to the theory of knot invariants and 3-manifold invariants which generalize the Jones polynomial and which may be considered as a mathematical version of the Witten invariants. Such a theory was introduced by N. Reshetikhin and the author on the ground of the theory of quantum groups. Here we use more general algebraic objects, specifically, ribbon and modular categories. Such categories in particular arise as the categories of representations of quantum groups. The notion of modular category, interesting in itself, is closely related to the notion of modular tensor category in the sense of G. Moore and N. Seiberg. For simplicity we restrict ourselves in this paper to the case of closed 3-manifolds.

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SORET, MARC, and MARINA VILLE. "SINGULARITY KNOTS OF MINIMAL SURFACES IN ℝ4." Journal of Knot Theory and Its Ramifications 20, no. 04 (April 2011): 513–46. http://dx.doi.org/10.1142/s0218216511009406.

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We study knots in 𝕊3 obtained by the intersection of a minimal surface in ℝ4 with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

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NAKANISHI, YASUTAKA, and YOSHIYUKI OHYAMA. "LOCAL MOVES AND GORDIAN COMPLEXES." Journal of Knot Theory and Its Ramifications 15, no. 09 (November 2006): 1215–24. http://dx.doi.org/10.1142/s0218216506005068.

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By the works of Gusarov [2] and Habiro [3], it is known that a local move called the Cnmove is strongly related to Vassiliev invariants of order less than n. The coefficient of the znterm in the Conway polynomial is known to be a Vassiliev invariant of order n. In this note, we will consider to what degree the relationship is strong with respect to Conway polynomial. Let K be a knot, and KCnthe set of knots obtained from a knot K by a single Cnmove. Let [Formula: see text] be the set of the Conway polynomials [Formula: see text] for a set of knots [Formula: see text]. Our main result is the following: There exists a pair of knots K1, K2such that ∇K1= ∇K2and [Formula: see text]. In other words, the CnGordian complex is not homogeneous with respect to Conway polynomial.

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Lee, I. J., and D. N. Yetter. "Dijkgraaf–Witten type invariants of Seifert surfaces in 3-manifolds." Journal of Knot Theory and Its Ramifications 26, no. 05 (April 2017): 1750026. http://dx.doi.org/10.1142/s0218216517500262.

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We introduce defects, with internal gauge symmetries, on a knot and Seifert surface to a knot into the combinatorial construction of finite gauge-group Dijkgraaf–Witten theory. The appropriate initial data for the construction are certain three object categories, with coefficients satisfying a partially degenerate cocycle condition.

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LEE, SANG YOUL, and MYOUNGSOO SEO. "FORMULAS FOR THE CASSON INVARIANT OF CERTAIN INTEGRAL HOMOLOGY 3-SPHERES." Journal of Knot Theory and Its Ramifications 18, no. 11 (November 2009): 1551–76. http://dx.doi.org/10.1142/s0218216509007610.

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In this paper, we introduce a representation of knots and links in S3 by integral matrices and then give an explicit formula for the Casson invariant for integral homology 3-spheres obtained from S3 by Dehn surgery along the knots and links represented by the integral matrices in which either all entries are even or the entries of each row are the same odd number. As applications, we study the preimage of the Casson invariant for a given integer and also give formulas for the Casson invariants of some special classes of integral homology 3-spheres.

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OHTSUKI, TOMOTADA. "INVARIANTS OF KNOTS DERIVED FROM EQUIVARIANT LINKING MATRICES OF THEIR SURGERY PRESENTATIONS." International Journal of Mathematics 20, no. 07 (July 2009): 883–913. http://dx.doi.org/10.1142/s0129167x09005583.

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The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.

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Boden, Hans U., Andrew J. Nicas, and Lindsay White. "Alexander invariants of periodic virtual knots." Dissertationes Mathematicae 530 (2018): 1–59. http://dx.doi.org/10.4064/dm785-3-2018.

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Journal articles on the topic 'Invariants of knots and 3-manifolds'

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